Sample Problem

A large square chessboard has a piece on every single square. When we take away the chess pieces in a row and a column, we find that 45 chess pieces have been taken away. How many chess pieces were originally on the board?

  chess pieces


A grid formation is a square with multiple layers of smaller squares. For example, a 3 × 5 grid is shown below.

Some grids are hollow; there will be a smaller empty square inside. In this section, you will be asked questions about students standing in grids. It is important to realize that any row of a grid shares exactly one student with any column of the grid, so if there are 5 students total in a row and a column combined, then there will be 3 students in each row and 3 in each column and 1 will be in both.

In an n´n chessboard, there will be 2n - 1 squares in 1 row and 1 column combined, so 2n - 1 = 45 and n = 23. This means that there are 23´23 = 529 squares on the board, so there were originally 529 pieces on the chessboard.